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Maybe there’s something better, but Bf6 looks like it works. Naturally if hxg Qxg6+ mates. Otherwise:
Bf6 Qf6
gxh+ Kh8
Rg8+ and wins the queen
If the king goes to f7 instead Rf3 pinning the queen
Bf6 Rf6
gxh+ Kh8
Rg8 mate
If the king goes to f7 instead Qh4+ and queens
Bf6 Qg3+
Kg3 Rxf6
gxh+ Rxh7 or Kh8
Qxf6
1.Bf6 Qxf6 (1…Txf6 2.gxh7 Kf7 3.Tg7 Kf8 4.h8Q#) 2.gxh7+ Kh8 3.Tg8 Txg8 4.hxg8Q+ Kxg8 5.Qxf6 +-.
I made the original post about Bf6. I think the line with Qxf6 and Kf7 could use a little more explanation. After Rf3 black could play Qxf3 and if white recaptures then Ra8 stopping white’s pawn. But after Qxf3 white could interpolate Qxf8+, queen with check and then capture on f3.
gxh7 looks like it works except for the dang rook pinning
if Bf6 black can’t capture with the rook Rf8 because
if gxh7 double check
just working through things lol